Tetration facts for kids

Kids Encyclopedia Facts

Tetration is the hyperoperation which comes after exponentiation. $^{x}{y}$ means y exponentiated by itself, (x-1) times. List of first 4 natural number hyperoperations:

Addition

$a + n = a\!\underbrace{''{}^{\cdots}{}'}_n$

a succeeded n times.
Multiplication

$a \times n = \underbrace{a + a + \cdots + a}_n$

a added to itself, n times.
Exponentiation

$a^n = \underbrace{a \times a \times \cdots \times a}_n$

a multiplied by itself, n times.
Tetration

${^{n}a} = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_n$

Note (operator associativity): ${^{n}a} = \underbrace{(a^{(a^{(\cdot^{\cdot^{(a)...)}}}}}_n$

a exponentiated by itself, n-1 times.

The above example is read as "the nth tetration of a".

Example

For the example, addition is assumed.

${^{2}3} =$

${3^{3}} =$

${3 \times 3 \times 3} =$

${3 \times (3 + 3 + 3)} =$

${3 \times {9}} =$

${3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 = 9 + 9 + 9} =$

$27$

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Tetration by period

Tetration facts for kids

Example

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See also